Can These Vector Sets Span Their Indicated Spaces?

In summary, the conversation discusses determining if sets of vectors span a given space, using methods such as row reduction and determinants. The person is unsure of how to proceed and plans to ask their professor for clarification.
  • #1
testme
68
0

Homework Statement


Determine if the following sets of vectors span the indicated space

a) {[0 -6 -6], [8 -3 5], [-9 7 -2]}, ℝ3.
b) {[2 1 7 -2], [3 5 4 5], [4 -4 -3 -3], [-5 0 6 -4]}, ℝ4.

Homework Equations





The Attempt at a Solution


a) a[0 -6 -6] + b[8 -3 5] + c[-9 7 -2] = [x y z]
x = 8b - 9c
y = -6a -3b + 7c
z = -6a + 5b - 2c

I don't know where to go from here - I'm sure if I can figure that out I'll be able to do b as well.
 
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  • #2
Use row reduction.
 
  • #3
We haven't been taught that yet that's why I'm not sure what he wants us to do..

We've been talking about linear independence, bases, and dimension but I don't know how I can go back and check if it spans the space.
 
  • #4
Have you been taught determinants? It would be much faster, but row reduction will give you the answer nicely.
 
  • #5
No, we haven't really been taught how to do anything with matrices, except maybe adding matrices or multiplying matrices by a scalar.
 
  • #6
testme said:
No, we haven't really been taught how to do anything with matrices, except maybe adding matrices or multiplying matrices by a scalar.

Google row reduction. There's lots of stuff out there and it really only takes 20-30 mins to learn once you've seen a few examples done.

EDIT : Here's a great explanation with an example :
 
Last edited by a moderator:
  • #7
Hmm, well, that helps, I think I can figure it out from here and I'll ask my professor if there was another method he expected us to know
 

FAQ: Can These Vector Sets Span Their Indicated Spaces?

What does it mean for a set of vectors to span Rn?

When a set of vectors spans Rn, it means that every vector in the n-dimensional space can be written as a linear combination of the given set of vectors. In other words, the set of vectors can reach every point in the n-dimensional space.

How can I determine if a set of vectors spans Rn?

To determine if a set of vectors spans Rn, you can use the method of Gaussian elimination. By setting up a matrix with the given vectors as columns and applying row operations, you can check if the matrix has a pivot in every row. If it does, then the set of vectors spans Rn. Alternatively, you can also check if the dimension of the span of the vectors is equal to Rn.

What is the significance of a set of vectors spanning Rn?

A set of vectors spanning Rn is significant because it forms a basis for the n-dimensional space. This means that any vector in Rn can be uniquely represented as a linear combination of the set of vectors. Additionally, it allows us to solve systems of linear equations and perform other important calculations in the n-dimensional space.

Can a set of vectors span Rn if it contains only one vector?

No, a set of vectors must contain at least n vectors to span Rn. This is because a vector space with n dimensions requires at least n linearly independent vectors to span it. If a set contains fewer than n vectors, then it will not be able to reach every point in Rn.

Is it possible for a set of vectors to span Rn if they are not linearly independent?

No, a set of vectors must be linearly independent to span Rn. If the vectors are not linearly independent, then they are redundant and can be written as a linear combination of the other vectors in the set. This means that the set will not have enough unique vectors to reach every point in Rn.

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